Inverse problem for Albertson irregularity index

Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio

Inverse problem for Albertson irregularity index

Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio

Atom-bond-connectivity index of certain graphs

The ABC index is one of the most applicable topological graph indices and several properties of it has been studied already due to its extensive chemical applications. Several variants of it have also been defined and used for several reasons. In this paper, we calculate the atom-bond connectivity index of some derived graphs such as double graphs, subdivision graphs and complements of some standard graphs.Publisher's Versio

On certain topological indices of the derived graphs of subdivision graphs

The derived graph [G]† of a graph G is the graph having the same vertex set as G, with two vertices of [G]† being adjacent if and only if their distance in G is two. Topological indices are valuable in the study of QSAR/QSPR. There are numerous applications of graph theory in the field of structural chemistry. In this paper, we compute generalized Randi´c, general Zagreb, general sum-connectivity, ABC, GA, ABC4, and GA5 indices of the derived graphs of subdivision graphs.Publisher's Versio

Maximum and minimum degree energies of p-splitting and p-shadow graphs

Let vi and vj be two vertices of a graph G. The maximum degree matrix of G is given in [2] by dij = {max {di, dj} if vi and vj are adjacent 0 otherwise. Similarly the (i, j)-th entry of the minimum degree matrix is defined by taking the minimum degree instead of the maximum degree above, [1]. In this paper, we have elucidated a relation between maximum degree energy of p−shadow graphs with the maximum degree energy of its underlying graph. Similarly, a relation has been derived for minimum degree energy also. We disprove the results EM(S0 (G)) = 2EM(G) and Em(S0 (G)) = 2Em(G) given by Zheng-Qing Chu et al. [3] by giving some counterexamples.Publisher's Versio

A new graph based on the semi-direct product of some monoids

In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics

Automorphisms of compact klein surfaces

Bu tezde kompakt Klein yüzeylerin otomorfizmleri teorisi incelendi. Klein yüzey dendiğinde yönlendirilebilen ya da yönlendirilemeyen bir Riemann yüzeyini anlayacağız. X, üzerindeki bir dianalitik yapı ile verilen bir Klein yüzey olsun. Dianalitik olan f : X -> X topolojik eşyapı dönüşümüne X in bir otomorfizmi denir. Hurwitz, cinsi g≥2 olan kenarsız yönlendirilebilir kompakt Klein yüzeyleri inceleyerek bu yüzeylerin otomorfizm gruplarının sonlu olduğunu ve aslında 84(g-1) i geçemeyeceğini göstermiştir. Macbeath de bu sınırın sonsuz çoklukta g değeri için elde edildiğini bulmuştur. Yüzyılımızda kompakt Klein yüzeyler ve bunların otomorfizmleri hâlâ önemli bir araştırma konusudur. Klein yüzeylerin otomorfizm grupları NEC gruplar yardımı ile çalışılabilir. Bu sebepten bu tezin birinci bölümünde NEC grupların genel özelliklerini belirttik. İkinci bölüm Klein yüzeyler teorisine ayrılmıştır. Üçüncü bölüm bu tezin en geniş bölümünü oluşturmaktadır. Bu son bölümde kenarlı ya da kenarsız, kompakt Klein yüzeylerin otomorfizm grupları incelenmiştir. Bu tezde konuyla ilgili birçok önemli teorem ve sonucu biraraya getirdiğimizi ümit ediyoruz.In this thesis the theory of automorphisms of compact Klein Surfaces is discussed. By a Klein Surface we mean a Riemann Surface which is orientable or non-orientable. Let X be a Klein Surface together with a dianalytic structure on X. A homeomorphism f : X -> X that is dianalytic will be called an auto morphism of X. Considering the orientable compact Klein Surface without boundary Hurwitz showed that for g ≥ 2, the groups of automorphisms of surfaces of genus g are finite, in fact, do not exceed 84(g-1), and Macbeath has shown that this bound is attained for infinitely many values of g. Compact Klein Surfaces and their automorphisms are still important research area in this century. Automorphism groups of Kleinian Surface can be studied with NEC groups. This is the reason that in chapter 1 of this thesis we worked out the general properties of NEC groups. The second chapter is devoted to the theory of Klein Surfaces. The chapter three is the largest section of the thesis. In this last chapter we studied out the automorphism groups in the case of compact Klein Surfaces with or without boundary. We beliew that we have collected a large number of important theorems and results on this topics